Let us give a brief account of the literature. M.-F. Bidault-Véron and L. Véron proved Inequality (REF ) in [1]} as a consequence of a rigidity result. Another proof by W. Beckner in [2]} relies on spectral methods, the Funk-Hecke formula and a duality result introduced by E. Lieb in [3]}. An earlier version corresponding to the range \(p\in (2,2^\#)\)  with \(2^\#=(2\,d^2+1)/(d-1)^2\)  if \(d\ge 2\)  and \(2^\#=+\infty \)  if \(d=1\)  was established by D. Bakry and M. Emery in [4]}, [5]}, using the carré du champ method and the heat flow. We refer to [6]} for a general overview of the carré du champ method in the context of Markov processes or linear diffusion equations, and to [7]}, [8]} for the extension to nonlinear diffusion equations. Also see Appendix  for further references in the case of the sphere.
